Level 1- Vice Grip
Level 6- Charge
Level 11- ThunderShock
Level 16- Gear Grind
Level 21- Bind
Level 26- Charge Beam
Level 31- Autotomize
Level 36- Mirror Shot
Level 39- Screech
Level 42- Discharge
Level 45- Metal Sound
Level 48- Shift Gear
Level 51- Lock-On
Level 54- Zap Cannon
Level 57- Hyper Beam
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it can learn Grass and Psychic moves
No.Magikarp cannot learn any TM moves if it evolves into gyarados then gyarados can learn TM moves and HM moves too. I Hope i helped you!
No Pokemon can learn more than four moves.
Magikarp can't learn any TM moves.
Alakazam cannot learn any HM moves.